The Stacks project

Definition 35.20.1. Germs of schemes.

1. A pair $(X, x)$ consisting of a scheme $X$ and a point $x \in X$ is called the germ of $X$ at $x$.

2. A morphism of germs $f : (X, x) \to (S, s)$ is an equivalence class of morphisms of schemes $f : U \to S$ with $f(x) = s$ where $U \subset X$ is an open neighbourhood of $x$. Two such $f$, $f'$ are said to be equivalent if and only if $f$ and $f'$ agree in some open neighbourhood of $x$.

3. We define the composition of morphisms of germs by composing representatives (this is well defined).

Definition 35.20.2. Let $f : (X, x) \to (S, s)$ be a morphism of germs. We say $f$ is étale (resp. smooth) if there exists a representative $f : U \to S$ of $f$ which is an étale morphism (resp. a smooth morphism) of schemes.